Le Monde puzzle [#1083]

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A Le Monde mathematical puzzle that seems hard to solve without the backup of a computer (and just simple enough to code on a flight to Montpellier):

Given the number N=2,019, find a decomposition of N as a sum of non-trivial powers of integers such that (a) the number of integers in the sum is maximal or (b) all powers are equal to 4.  Is it possible to write N as a sum of two powers?

It is straightforward to identify all possible terms in these sums by listing all powers of integers less than N

pool=(1:trunc(sqrt(2019)))^2
for (pow in 3:11)
  pool=unique(c(pool,(2:trunc(2019^(1/pow)))^pow))

which leads to 57 distinct powers. Sampling at random from this collection at random produces a sum of 21 perfect powers:

 1+4+8+9+16+25+27+32+36+49+64+81+100+121+125+128+144+169+196+243+441

But looking at the 22 smallest numbers in the pool of powers leads to 2019, which is a sure answer. Restricting the terms to powers of 4 leads to the sequence

1⁴+2⁴+3⁴+5⁴+6⁴ = 2019

And starting from the pools of all possible powers in a decomposition of 2019 as the sum of two powers shows this is impossible.

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This article is originally published at https://xianblog.wordpress.com
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